Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 022, 13 pages
G-Invariant Deformations of Almost-Coupling
Poisson Structures
Jose´ Antonio VALLEJO † and Yury VOROBIEV ‡
† Facultad de Ciencias, Universidad Auto´noma de San Luis Potos´ı, Me´xico
E-mail: jvallejo@fc.uaslp.mx
URL: http://galia.fc.uaslp.mx/~jvallejo/
‡ Departamento de Matema´ticas, Universidad de Sonora, Me´xico
E-mail: yurimv@guaymas.uson.mx
Received October 31, 2016, in final form March 28, 2017; Published online April 02, 2017
https://doi.org/10.3842/SIGMA.2017.022
Abstract. On a foliated manifold equipped with an action of a compact Lie group G, we
study a class of almost-coupling Poisson and Dirac structures, in the context of deformation
theory and the method of averaging.
Key words: Poisson geometry; Dirac structures; deformation; averaging
2010 Mathematics Subject Classification: 53D17; 70G45; 58H15
1 Introduction
In this paper, we develop further the results of [18, 19], on the construction of invariant Poisson
and Dirac structures via the averaging method on foliated Poisson manifolds with symmetry in
the context of deformation theory.
For a Poisson bivector field Π on a foliated manifold (M,F), the F-almost-coupling proper-
ty [16] means the existence of a normal bundle structure H of F , such that in the H-dependent
bigraded decomposition of Π, the mixed term of bidegree (1, 1) vanishes. In particular, such
class of Poisson structure contains the F-coupling Poisson structures which play important roles
in some problems of semi-local Poisson geometry [9, 10, 16, 20]. Moreover, the (almost) coupling
constructions can be naturally extended to the Dirac category [5, 17, 21].
Now, starting with a foliated Poisson manifold (M,F , P ), equipped with a leaf preserving
action of a compact connected Lie group G, our point is to study, in the context of the averag-
ing method, some deformations {Πε}ε∈[0,1] of the leaf-tangent
1 Poisson bivector field P in the
class of the F-almost-coupling Poisson structures Πε. Our approach is based on the averaging
technique [18] related to a class of exact gauge transformations for Poisson and Dirac structures.
The idea of the construction of G-invariant Poisson structures is to follow the path:
Poisson
averaging
−−−−−−→ G-invariant Dirac
non-degeneracy
−−−−−−−−−→ G-invariant Poisson.
For the case of a G-action which is locally Hamiltonian relative to P , we give some results
about the realization of the above scheme for a class of F-almost-coupling Poisson deformations
of P . Within the framework of perturbation theory, these results can be applied to the study of
invariant normal forms for Hamiltonian systems of adiabatic type [1], associated to deformations
of Poisson structures [2, 4].
This paper is a contribution to the Special Issue “Gone Fishing”. The full collection is available at
http://www.emis.de/journals/SIGMA/gone-fishing2016.html
1Following [17], a leaf-tangent (Poisson) bivector field on a foliated manifold is a section of
∧2 TF .
2 J.A. Vallejo and Yu. Vorobiev
In a generalized setting, by a Hamiltonian system of adiabatic type on a Poisson foliation
(M,F , P ) with G-symmetry, we mean a Hamiltonian system relative to a deformed Poisson
structure Πε = P +εΛ and a function F ∈ C∞(M). The deformation of P is given by a bivector
field Λ ∈ Γ(∧2H), where H ⊂ TM is a normal bundle of the foliation F . In the (adiabatic) limit
ε → 0, the unperturbed Hamiltonian system (M,P, F ) is G-invariant and describes the fast
dynamics along the leaves of F . The key point is to move the original perturbed system to a G-
invariant one by using a near-identity transformation. The construction of the corresponding
G-invariant model is related to the averaging procedure for the deformed Poisson structure Πε.
In a particular setting this problem was studied in [4], here we present a general mechanism.
2 Preliminaries
Let us recall some basic facts that will be used later on, related to the averaging procedure with
respect to the action of compact Lie groups, for Poisson and Dirac structures [18].
2.1 Gauge transformations of Dirac manifolds
Let (M,D) be a Dirac manifold, that is, a smooth regular distribution D ⊂ TM⊕T ∗M which is
maximally isotropic with respect to the natural symmetric pairing on TM ⊕ T ∗M , and involu-
tive with respect to the Courant bracket [7, 8]. The Dirac manifold (M,D) carries a (singular)
presymplectic foliation (S, ω): Its leaves are the maximal integral manifolds of the integrable (sin-
gular) distribution C := pT (D) ⊂ TM , and the leafwise presymplectic structure ω is defined by
ωq(X,Y ) = −α(Y ), for (X,Y ) ∈ Cq := C∩TqM and (X,α) ∈ Dq, q ∈M . Here pT : TM ⊕ T ∗M
→ TM is the natural projection.
We can modify the leafwise presymplectic structure ω by the pull back of a closed 2-form
on the base B ∈ Ω2(M): For each presymplectic leaf (S, ωS), we define the new presymplectic
structure as ωS + ι∗SB, where ιS : S ↪→M is the inclusion map. Then, the foliation S equipped
with the deformed leafwise presymplectic structure gives rise to the new Dirac structure
τB(D) = {(X,α− iXB) | (X,α) ∈ D}.
This transformation τB preserves the presymplectic foliation of D, and is called the gauge trans-
formation associated to the closed 2-form B [6, 14].
In particular, the foliation (S, ω) is symplectic if and only if D is the graph of a Poisson
bivector field Π on M ,
D = Graph Π =
{(
Π\α, α
)
|α ∈ Ω1(M)
}
, (2.1)
where Π\ : T ∗M → TM is the induced vector bundle endomorphism given by α 7→ iαΠ. Condi-
tion (2.1) can be expressed as follows
D ∩ (TM ⊕ {0}) = {(0, 0)}.
Notice that, in general, for a given closed 2-form B, the gauge transformation τB takes Graph Π
to another Dirac structure
τB(Graph Π) =
{(
Π\α, α− iΠ\αB
)
|α ∈ Ω1(M)
}
,
which may not necessarily come from a Poisson bivector field. We have the simple, but useful,
criterion (see [14, pp. 5–6]).
G-Invariant Deformations of Almost-Coupling Poisson Structures 3
Lemma 2.1. If the endomorphism
(
Id−B[ ◦Π\
)
: T ∗M → T ∗M
is invertible, then the Dirac structure τB(GraphΠ) is the graph of the Poisson tensor τB(Π)
whose induced endomorphism is defined by
τB(Π)
] = Π] ◦
(
Id−B[ ◦Π]
)−1
.
Here, we denote by B[ : TM → T ∗M the vector bundle endomorphism given by X 7→ iXB.
2.2 G-averaging procedure
Let G be a connected, compact Lie group, and g its Lie algebra. Suppose we are given a smooth
(left) action Φ: G×M →M on a manifold M . For every a ∈ g, the corresponding infinitesimal
generator is denoted by aM ∈ X(M),
aM (q) :=
d
dt
∣
∣
∣
∣
t=0
Φexp(ta)(q), q ∈M.
For every tensor field T on M , we denote by 〈T 〉G its G-average, which is defined as
〈T 〉G :=
∫
G
Φ∗gT dg,
where dg is the normalized Haar measure on G. This averaging procedure can be also applied
to Dirac structures [18]: Let D ⊂ TM ⊕ T ∗M be a Dirac structure on M , and let (S, ω) be its
associated presymplectic foliation, carrying the leafwise presymplectic form ω.
Definition 2.2. The G-action on M is compatible with the Dirac structure D if each leaf S,
of S, is invariant under the action of G, and there exists a R-linear mapping ρ ∈ Hom(g,Ω1(M))
such that
iaMωS = −ι
∗
Sρa, (2.2)
for every a ∈ g (here, ωS is a presymplectic structure on S, and ιS : S ↪→ M is the canonical
injection).
Equivalently, condition (2.2) can be rewritten as follows
(aM , ρa) ∈ Γ(D) for all a ∈ g.
Then, as a consequence of the fact that G is compact connected (hence the exponential map is
surjective) we have the following fact [18].
Lemma 2.3. If the G-action is compatible with D, then there exists a Dirac structure D on M
with the following properties:
(a) The leafwise presymplectic form of D is defined as
〈ω〉GS := ωS − ι
∗
SdΘ,
where Θ ∈ Ω1(M) is the 1-form determined in terms of the G-action and ρ by
Θ :=
∫
G
(∫ 1
0
Φ∗exp(τa)ρa dτ
)
dg, g = exp a. (2.3)
4 J.A. Vallejo and Yu. Vorobiev
(b) D is G-invariant:
(X,α) ∈ Γ(D) implies that (Φ∗gX,Φ
∗
gα) ∈ Γ(D) for all g ∈ G.
(c) The Dirac structure D is is related to D by an exact gauge transformation:
D = {(X,α+ iXdΘ) | (X,α) ∈ D}.
The Dirac structure D will be called the G-average of D relative to the compatible G-action.
As we have mentioned above, for the case in which the Dirac structure is the graph of a Poisson
tensor Π on M , D = Graph Π, its G-average D = τB(D), where B = −dΘ, does not necessarily
come from a Poisson structure. In other words: In general, the averaging of Poisson structures
via compatible G-actions only leads to G-invariant Dirac structures; but, by Lemma 2.1, in the
particular case of an invertible endomorphism
Id +(dΘ)[ ◦Π\,
we can say something more, namely, that the G-average D is the graph of the G-invariant
Poisson tensor Π = τB(Π).
2.3 G-invariant connections
Recall that a (generalized Ehresmann) connection on a manifold M [13], is a vector-valued 1-form
γ ∈ Ω1(M ;TM) satisfying the following conditions: γ2 = γ, and the rank of the distribution
Im γ ⊂ TM is constant on M (this last condition is a consequence of the first for M connected).
Assume that the action Φ: G × M → M , of the compact, connected Lie group G on the
connected manifold M , preserves the image of γ, dqΦg ◦γq = γΦg(q), for all q ∈M , g ∈ G. Then,
the G-average of γ is the vector-valued 1-form 〈γ〉G ∈ Ω1(M,TM) defined by the formula
〈γ〉G(X) :=
∫
G
Φ∗g(γ((Φg)∗X) dg,
where X ∈ X(M) is any vector field, and this is a G-invariant connection on M .
3 F-almost-coupling structures
In this section, we present some basic definitions and facts concerning the coupling procedure
on foliated Poisson and Dirac manifolds (for more details, see [16, 17, 20]). The term ‘coupling’
comes from Sternberg’s coupling method on symplectic fiber bundles (see, for example, [12]).
Suppose we are given a manifold with a (regular) foliation (M,F). We will denote by V := TF
the tangent bundle of F , and by V0 = Ann(TF) ⊂ T ∗M its annihilator. By a normal bundle
of the foliation F we mean a sub-bundle H ⊂ TM which is complementary to the tangent
bundle V,
TM = H⊕ V. (3.1)
To a normal bundle H, there is associated a vector-valued 1-form γ ∈ Ω1(M ;TM), defined as the
canonical projection along H, γ := prH : TM → V. This form satisfies γ
2 = γ and Im γ = V, and
hence defines an Ehresmann connection on the foliated manifold (M,F) (see [13]). The curvature
of the connection γ is the vector-valued 2-form Rγ ∈ Ω2(M ;V) given by Rγ = 12 [γ, γ]FN, where
[·, ·]FN denotes the Fro¨licher–Nijenhuis bracket for vector valued forms on M [13]. The curvature
G-Invariant Deformations of Almost-Coupling Poisson Structures 5
controls the integrability of the normal sub-bundle, in the sense that H is integrable if and only
if γ is flat, Rγ = 0.
According to (3.1), we have the dual splitting
T ∗M = V0 ⊕H0. (3.2)
Then, (3.1) and (3.2) induce an H-dependent bigrading of multivector fields and differential
forms on M . For any A ∈ Γ(∧kTM) and α ∈ Ωk(M), we have A =
∑
s+l=k
As,l and α =
∑
s+l=k
αs,l, where the elements As,l and αs,l, belonging to the subspaces Γ(∧sH) ⊗ Γ(∧lV) and
Γ(∧sV0) ⊗ Γ(∧lH0), respectively, are said to be multivector fields and forms of bidegree (s, l).
Moreover, the exterior differential d on M inherits the H-bigrading decomposition (see [16]) in
the form d = d1,0 + d2,−1 + d0,1.
We will need the following definition [16, 20].
Definition 3.1. A Poisson structure Π on the foliated manifold (M,F) is said to be F-almost-
coupling, via a normal bundle H of the foliation F , if the image of V0 under the vector bundle
morphism Π] : T ∗M → TM , is contained in H:
Π]
(
V0
)
⊆ H. (3.3)
This condition means that, in the bigraded decomposition of Π associated to (3.1), the mixed
term Π1,1 is zero and Π = Π2,0 + Π0,2, where Π2,0 ∈ Γ(∧2H) and Π0,2 ∈ Γ(∧2V) is a Poisson
tensor. The characteristic distribution of Π is contained into the (possibly non-integrable)
distribution H⊕Π]0,2(H
0).
Remark 3.2. Notice that conditions (3.1), (3.3) hold whenever the following transversality and
regularity conditions are satisfied (in this case we say that the bivector field Π is compatible
with the foliation F):
Π]
(
V0
)
∩ V = {0}, (3.4)
and
rank Π]
(
V0
)
= constant on M. (3.5)
Then, it follows from (3.4), (3.5) that a normal bundle of F in (3.3) can be constructed as
follows: H = H′ ⊕ Π](V0), where H′ ⊂ TM is an arbitrary sub-bundle, complementary to the
regular distribution Π]
(
V0
)
⊕ V.
The F-coupling situation occurs when, along with (3.4), we have
rank Π]
(
V0
)
= codimF on M,
and hence the normal bundle H of F in (3.3), associated to Π, is unique and given by
H = Π]
(
V0
)
. (3.6)
In this case, Π is said to be an F-coupling Poisson structure. The factorization of the Jacobi
identity for Π (see [18, equations (6.4)–(6.6)]) implies that the intrinsic connection γ, associated
with the normal bundle (3.6), possesses the following properties [16, 20]: The connection γ is
Poisson on the Poisson bundle (M,F ,Π0,2), that is, LXΠ0,2 = 0 for any projectable section
X ∈ Γpr(H) (recall that the projectability property for X on the foliated manifold (M,F) is
6 J.A. Vallejo and Yu. Vorobiev
expressed as [X,Γ(V)] ⊂ Γ(V)). Moreover, the curvature of γ takes values in the space of
Hamiltonian vector fields of Π0,2:
Rγ(X,Y ) = −Π]0,2dσ(X,Y ) for all X,Y ∈ Γpr(H). (3.7)
Here, the 2-form σ ∈ Γ(∧2V0), called the coupling form, is uniquely determined by Π, and
satisfies the γ-covariant constancy condition d1,0σ = 0. Indeed, we can write the following
explicit expression in terms of the horizontal part of Π and the horizontal lifting induced by γ [20]:
σ(u1, u2) = −
〈(
Π]2,0
∣
∣
V0
)−1
hor(u1),hor(u2)
〉
,
for any u1, u2 ∈ X (M) vector fields on M .
The notion of F-almost-coupling structures can be naturally generalized to the Dirac set-
ting [17]. Given a Dirac structure D ⊂ TM ⊕ T ∗M on the foliated manifold (M,F), we define
the tangent distribution
H(D,F) :=
{
X ∈ TM | ∃α ∈ V0 such that (X,α) ∈ D
}
.
On the other hand, fixing a normal bundle H of F , we consider the distributions
DH := D ∩
(
H⊕ V0
)
and DV := D ∩
(
V⊕H0
)
.
In the general case, these are singular distributions, but the almost-coupling hypothesis implies
that their respective ranks are constant, rankDH = rankH and rankDV = rankV.
Definition 3.3. D is said to be an F-almost-coupling Dirac structure, via a normal bundle H
of F , if
H(D,F) ⊆ H. (3.8)
One can show that condition (3.8) is equivalent to the following one:
D = DH ⊕DV .
Also, one says that D is an F-coupling Dirac structure [17, 21] if H(D,F) is a normal bundle
of F ,
TM = H(D,F)⊕ V.
Notice that (3.8) implies the following property:
DV ∩ (V⊕ {0}) = {0}.
It follows from here that a given an F-almost coupling Dirac structure D via H induces a
leaf-tangent Poisson bivector field P ∈ Γ(∧2V) such that
DV =
{
(P ]η, η) | η ∈ H0
}
. (3.9)
Remark 3.4. It easy to see that these definitions agree with the corresponding notions in the
Poisson case. Indeed, if D = Graph Π, for a certain Poisson tensor Π, then H(D,F) = Π](V0).
The relation between geometric data and F-coupling Dirac structures [17, 18, 21], carries
over to the almost-coupling setting, with some particularities: given a connection γ, the almost-
coupling Dirac structure directly induces a leaf-tangent Poisson structure P , and no coupling
form σ (in the terminology of [18]) appears. We can use that correspondence to characterize (in
G-Invariant Deformations of Almost-Coupling Poisson Structures 7
terms of P ) a particular class of gauge transformations which preserve2 the class of F-almost
coupling Dirac structures on a given foliated manifold (M,F) (this class naturally appears in
the context of the averaging method). To this end, let us pick an arbitrary 1-form Q ∈ Γ(V0)
and consider the exact gauge transformation
D 7→ D := τB(D), B = −dQ. (3.10)
Then, relative to decomposition (3.2), we have B = B2,0 + B1,1, where B2,0 = d1,0Q and
B1,1 = d0,1Q.
Lemma 3.5. Let D be an F-almost coupling Dirac structure via a normal bundle H of F .
Then, the exact gauge transformation (3.10) maps D into the Dirac structure D which is again
F-almost coupling, this time via the normal bundle
H :=
(
Id +P ] ◦B[1,1
)
(H).
Here, P ∈ Γ(
∧2 V) is the Poisson bivector field in (3.9).
Proof. For an arbitrary closed 2-form B = B2,0 +B1,1 +B0,2, the gauge transformation τB(D)
is a Dirac structure consisting of elements of the form
(X1,0 +X0,1)⊕
((
α1,0 −B
[
2,0X1,0 −B
[
1,1X0,1
)
+
(
α0,1 −B
[
1,1X1,0 −B
[
0,2X0,1
))
,
where (X1,0 +X0,1, α1,0+ α0,1) ∈ D. If B0,2 = 0, then taking into account (3.9), we can describe
the characteristic distribution of τB(D) as
H(τB(D),F) =
{(
Id +P ] ◦B[1,1
)
X |X ∈ H(D,F)
}
.
This proves the statement. 
Notice also that the exact gauge transformation (3.10) leaves invariant the set of all F-
coupling Dirac structures on (M,F) [18].
4 The averaging theorems for deformations
Let (M,F , P ) be a Poisson foliation, consisting of a regular foliation F on M , and a leaf-tangent
Poisson bivector field P ∈ Γ(∧2V). Recall that we denote by V = TF ⊂ TM the tangent bundle
of F and by V0 ⊂ T ∗M its annihilator. The Poisson tensor P is characterized by the property
that the symplectic leaf of P through each point q ∈ M , is contained into the leaf Fq of the
regular foliation. Another characterization, given in [17], states that the leaves of F are Poisson
submanifolds.
Consider a smooth (left) action on M , Φ: G × M → M , of a connected and compact
Lie group G, which is compatible with the Poisson structure P ∈ Γ(∧2V), in the sense of
Definition 2.2: For every a ∈ g, the infinitesimal generator aM has the form
aM = P
]µa for all a ∈ g, (4.1)
for a certain R-linear mapping µ ∈ Hom(g,Ω1(M)). In particular, this property means that the
G-action preserves the leaves of the symplectic foliation of P , and condition (2.2) holds, that is,
the G-action is compatible with the associated Dirac structure D = GraphP .
2In general, arbitrary gauge transformations do not carry almost-coupling Dirac structures into almost-coupling
structures again. This does happen for the gauge transformations (3.10), however.
8 J.A. Vallejo and Yu. Vorobiev
A G-action on the Poisson foliation is said to be locally Hamiltonian, if each infinitesimal
generator aM is a locally Hamiltonian vector field relative to P , in other words, one can choose
the 1-form µa in (4.1) to be closed on M ,
µa ∈ Ω
1
cl(M) for all a ∈ g. (4.2)
Suppose now that we start with a smooth deformation {Πε}ε∈[0,1] of the leaf-tangent Poisson
structure P , so each Πε is a Poisson tensor on M , the whole family is smoothly dependent in ε,
and is such that Π0 = P . Moreover, we assume that, for each ε ∈ [0, 1], the bivector field Πε is
an F-almost-coupling Poisson structure via an ε-independent normal bundle H of F :
Π]ε
(
V0
)
⊆ H for all ε ∈ [0, 1]. (4.3)
Such a family {Πε}, will be called an F-almost-coupling Poisson deformation of P via H. It
follows from (4.3) that each Πε admits the following bigraded decomposition relative to (3.1):
Πε = (Πε)2,0 +(Πε)0,2, where (Π0)0,2 = P . We will also assume that the leaf-tangent component
of Πε of bidegree (0, 2) is independent of ε and hence
(Πε)0,2 = P for all ε ∈ [0, 1].
Then, (Πε)2,0 = εΛε for a certain bivector field Λε of bidegree (2, 0), smoothly varying in ε, and
therefore, the deformation of P can be parameterized as
Πε = P + εΛε, Λε ∈ Γ
(
∧2H
)
. (4.4)
For a fixed R-linear mapping µ ∈ Hom(g,Ω1(M)) in (4.1), we can decompose µ = µ1,0 +µ0,1
relative to splitting (3.2), associated to a fixed normal bundle H in (4.3).
Finally, let us associate to the family {Πε} (as in (4.4)) a smooth ε-dependent family of Dirac
structures
Dε := Graph Πε = GraphP ⊕Graph εΛε, (4.5)
which is to be viewed as a deformation of the ‘limiting’ Dirac structure D0 = Graph Π0 =
GraphP . Then, for each ε, the Dirac structure Dε also satisfies the F-almost-coupling condi-
tion (3.8). Under the above hypotheses, we are in position to state our main results regarding
the existence of the G-average for these deformations of Dirac and Poisson structures. The first
one is a consequence of Lemma 2.3.
Theorem 4.1. For every ε ∈ [0, 1], the G-average Dε, of the F-almost-coupling Dirac struc-
ture Dε in (4.5), is well-defined and given by the exact gauge transformation
Dε = {(X,α+ iXdΘ) | (X,α) ∈ Dε}, (4.6)
where Θ is the 1-form defined by (2.3), with ρ = µ0,1.
Proof. It follows from (4.4) that, for every ε ∈ [0, 1] and a ∈ g, we have
Π\ε(µa)0,1 = (Πε)
\
0,2(µa)0,1 = (Πε)
\
0,2µa = P
\µa = aM ,
and, hence, the G-action preserves the symplectic leaves of Π\ε, and the compatibility condi-
tion (2.2) holds for ρ = µ0,1. Then, by Lemma 2.3, the G-average of Dε is well-defined and
given by (4.6). 
G-Invariant Deformations of Almost-Coupling Poisson Structures 9
Theorem 4.2. If the G-action is locally Hamiltonian (condition (4.2)), then, for any G-
invariant open connected set with compact closure, N ⊂M , and for sufficiently small ε ∈ ]0, 1[,
the restriction of the Dirac structure (4.6), Dε|N , is the graph of a G-invariant Poisson struc-
ture Πε on N defined by
Π
]
ε = Π
]
ε ◦
(
Id +(dQ)[ ◦Π]ε
)−1
(4.7)
with Π0 = P . Here Q ∈ Γ(V0) is expressed as
Q := −
∫
G
(∫ 1
0
Φ∗exp(τa)(µa)1,0 dτ
)
dg, g = exp a. (4.8)
Moreover, the Poisson structure Πε is F-almost-coupling via the following G-invariant normal
bundle of F :
H := Span
{
X¯ = X + P ]dQ(X) |X ∈ Γpr(H)
}
. (4.9)
Proof. Under the hypothesis that the G-action is locally Hamiltonian, let us fix a G-invariant
relatively compact domain N ⊆ M . Let B = −dΘ, with Θ constructed as in (2.3). By
Lemma 2.1 it suffices to show that there exists a 0 < δ < 1 such that
(
Id−B[ ◦Π\ε
)
is invertible on N for ε ∈ [0, δ]. (4.10)
In terms of the bigraded components of µ, the closedness condition dµ = 0 splits into
d0,1µ0,1 = 0, d1,0µ0,1 = −d0,1µ1,0, d2,−1µ0,1 = −d1,0µ1,0.
By using these relations, we see from (2.3) that B = −dΘ = −dQ, where Q ∈ Γ(V0) is given
by (4.8). It follows that B = B2,0 +B1,1, where B2,0 = −d1,0Q and B1,1 = −d0,1Q. From here,
taking into account (4.4), for an arbitrary α = α1,0 + α0,1 ∈ Ω1(M), we get
(
B[ ◦Π\ε
)
(α1,0 + α0,1) = B
[
1,1 ◦ P
\α0,1 + ε
(
B[2,0 ◦ (Λε)
\
2,0α1,0 +B
[
1,1 ◦ (Λε)
\
2,0α1,0
)
.
This shows that the matrix of the morphism Id−B[ ◦ Π\ε : T ∗M → T ∗M , in a local basis
compatible with the splitting (3.2), has the form
(
I ∗
0 I
)
+O(ε).
This fact, together with a standard compactness argument, proves (4.10). Furthermore, it
follows that
(
Id +(dQ)[ ◦ P ]
)−1
= Id−(dQ)[ ◦ P ], (4.11)
and
P ] ◦ (dQ)[ ◦ P ] = 0. (4.12)
From here and (4.7), we conclude that Π0 = P . Finally, to show that the Poisson structure Πε
is F-almost coupling, we consider the connection γ associated to the normal bundle H and its
G-average 〈γ〉G. Then, the normal bundle ker〈γ〉G just coincides with H in (4.9) (see [18]).
Applying Lemma 3.5 ends the proof. 
10 J.A. Vallejo and Yu. Vorobiev
Consider now the case of Πε a coupling Poisson structure for ε 6= 0, that is, H = Π
]
ε(V0) is
a normal bundle of F . Then, the corresponding Dirac structure is represented as
Dε = Graph(Πε) =
{(
εX + P ]α, α− iXσε
)
|X ∈ Γ(H), α ∈ Γ
(
H0
)}
,
where the 2-form σε ∈ Γ(∧2V0) smoothly depends in ε, satisfies the H-covariant constancy
condition d1,0σε = 0, and has the following expansion around ε = 0:
σε = c+ εσ +O
(
ε2
)
.
Here, the 2-form c ∈ Γ(∧2V0) takes values in the Casimir functions of P ,
c(X,Y ) ∈ Casim(M,P ) for all X,Y ∈ Γpr(H),
and the 2-form σ ∈ Γ(∧2V0) satisfies the curvature identity (3.7) with Π0,2 = P (for more details,
see [16, 18]). From these remarks and Theorem 4.2, we deduce the following consequence.
Corollary 4.3. In the coupling case, for a locally Hamiltonian G-action, the G-average Dε is
an F-coupling Dirac structure via the invariant normal bundle H¯ given by
Dε =
{(
εX + P ]α, α− iXσε + iεX+P ]αdQ
)
|X ∈ Γ(H), α ∈ Γ
(
H0
)}
.
In particular, D0 = Graph(P ).
A natural question concerns the relationship between the original, ε-dependent, Poisson struc-
ture Πε, and its G-average Πε, as ε tends to 0.
Theorem 4.4. Let Πε be the G-average of the F-almost-coupling Poison structure Πε defined
in (4.7). Let N ⊂ M be a G-invariant open subset whose closure is compact. There exists
a smooth isotopy φε : N →M , such that, for ε sufficiently small,
φ∗εΠε = Πε, φ0 = Id
on N .
Proof. Following the reasoning in the proof of Theorem 4.2, we can show that there exists a δ >
0 such that the gauge transformation of Πε, associated to the exact 2-form tB = −tdQ, exists
for all ε ∈ [0, δ], t ∈ [0, 1], and gives the 2-parameter family of Poisson structures on N , Πε,t,
characterized by
Π]ε,t = Π
]
ε ◦
(
Id +t(dQ)[ ◦Π]ε
)−1
. (4.13)
Then, Πε,0 = Πε and Πε,1 = Πε. Fixing ε ∈ [0, δ], one can verify [11, 18] that the time-dependent
vector field on N given by
Zε,t = −Π
]
ε,t(Q) = −Π
]
ε ◦
(
Id +t(dQ)[ ◦Π]ε
)−1
(Q) (4.14)
satisfies the homotopy equation (where [[·, ·]] denotes the Schouten bracket for multivector fields
on M [15])
[[Zε,t,Πε,t]] = −
dΠε,t
dt
. (4.15)
Moreover, we have Π0,t = P and Z0,t = −P ](Q) = 0. These facts, together with the compactness
of the closure of N , show that (by shrinking δ > 0 if necessary), for each ε ∈ [0, δ], the flow FltZε,t
of Zε,t is well-defined on N , for all t ∈ [0, 1]. Then, it suffices to put φε = FltZε,t |t=1. 
G-Invariant Deformations of Almost-Coupling Poisson Structures 11
5 Infinitesimal deformations
In this section, we will derive a first-order approximation formula for the averaged Poisson
structure in (4.7).
Let Πε = P + εΛ0 + O(ε2) be a family of almost-coupling Poisson structures on M . From
the Jacobi identity [[Πε,Πε]] = 0 = [[P, P ]], and it follows that the bivector field Λ0 ∈ Γ(∧2H)
is a 2-cocycle in the Lichnerowicz–Poisson complex of (M,P ), [[P,Λ0]] = 0. This 2-cocycle Λ0,
determines the infinitesimal (first-order) part of the almost-coupling deformation of P . Then,
taking into account the identities (4.11) and (4.12), we deduce from (4.7)
Πε = P + εΛ0 +O
(
ε2
)
, (5.1)
where Λ0 is another G-invariant 2-cocycle in the Lichnerowicz–Poisson complex, given by
Λ
\
0 =
(
Id−P ] ◦ (dQ)[
)
◦ Λ\0 ◦
(
Id−(dQ)[ ◦ P ]
)
.
By varying (5.1) with respect to ε, we get the following infinitesimal version of Theorem 4.4.
Proposition 5.1. The cohomology classes of the 2-cocycles Λ0 and Λ0, coincide.
Proof. For the infinitesimal generator Zε,t in (4.14), and the family of Poisson structures Πε,t
in (4.13), we can evaluate their expansion around ε = 0: Zε,t = εWt + O(ε2) and Πε,t =
P + εΨt + O(ε2), where Wt and Ψt are a time-dependent vector field and a bivector field,
respectively. Putting these expansions into (4.15), and collecting all first-order terms in ε, leads
to the equation [[Wt, P ]] = −dΨtdt . Integrating this equation with respect to t, and taking into
account that Ψ0 = Λ0, and Ψ1 = Λ0, we get Λ0 − Λ0 = [[w,P ]], where w = −
∫ 1
0 Wt dt. 
Remark 5.2. These results can be used to construct normal forms for perturbed dynamics
associated to almost-coupling deformations of foliated Poisson manifolds with symmetry. Typi-
cally, such perturbed dynamics appear in the context of adiabatic theory [1] on nontrivial phase
spaces, particularly those in which no global action-angle coordinates can be introduced [2, 3, 4].
We have a simple example of an F-almost-coupling Poisson deformation when the foliation F
is a fibration.
Example 5.3. Suppose we start with a Poisson fiber bundle (pi : M → S, P ) whose base S
carries a Poisson structure ψ ∈ Γ(∧2TS). Assume that there exists a flat Poisson connection γ
on M , associated to an integrable distribution H, which is complementary to the vertical one
V = ker dpi. Let horγ(ψ) ∈ Γ(∧2H) be the γ-horizontal lift of the Poisson bivector field ψ.
Then, by using the standard properties of the Schouten bracket (see [15]), one can show that
the bivector field
Πε = P + εhor
γ(ψ) (5.2)
satisfies the Jacobi identity and gives an almost-coupling Poisson structure for every ε ∈ [0, 1].
It is clear that Πε is a coupling Poisson tensor on M if and only if the bivector field ψ is
non-degenerate, and hence induces a symplectic form on S.
Finally, let us say some words about the physical meaning of the preceding constructions.
Consider the Hamiltonian vector field X(ε) = idFP + εidF hor
γ(ψ), relative to a function
F ∈ C∞(M) and the Poisson structure (5.2). As stated in the Introduction, the corresponding
dynamical system
ξ˙ = εidF hor
γ(ψ), x˙ = idFP, (5.3)
12 J.A. Vallejo and Yu. Vorobiev
belongs to the class of the so-called slow-fast Hamiltonian systems [1], where the coordinates
ξ = (ξi), along the base S, and the coordinates x = (xα), along the fibers of pi, are called the
slow and fast variables, respectively. We observe that the infinitesimal generator of the Poisson
isotopy φε in Theorem 4.4, controls the ‘geometric’ part of an invariant normalization transfor-
mation for the system (5.3) as ε tends to zero. As stated in Remark 5.2, these transformations,
and the normal forms to which they lead, are crucial for determining asymptotic properties
of the system, such as stability, existence of periodic orbits, or the computation of adiabatic
invariants, see [2] for some examples.
Acknowledgements
We express our gratitude to the anonymous referees, whose detailed comments and criticism
have greatly improved the contents of this paper. Also, we thank the organizers of the Gone
Fishing Meeting at Berkeley (2015), for the opportunity of presenting and discussing some of
the results exposed here. This work was partially supported by the Mexican National Council
of Science and Technology (CONACyT), under research projects CB-2012-179115 (JAV) and
CB-2013-219631 (YuV).
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